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Title: Right matrix fraction description for stochastically singular models: Structure theory and estimation Authors:  Juho Koistinen - University of Helsinki (Finland) [presenting]
Bernd Funovits - University of Helsinki (Finland)
Abstract: The focus is on the estimation of stochastic processes with rational spectral densities that are rank deficient. Interest in these processes has emerged in relation to generalized dynamic factor models (GDFMs), which contain fewer economic shocks than endogenous variables. Our contribution is to propose a right matrix fraction description (MFD) realization of the transfer function, $\chi_{t}=k(z)\varepsilon_{t}=d(z)c(z)^{-1}\varepsilon_{t}$ with $d(z)\in\mathbb{R}^{N\times q}$ and $c(z)\in\mathbb{R}^{q\times q}$ which has two advantages over the usual two-step estimation procedure (where firstly a static transformation reduces to a static factor process with full rank covariance matrix and secondly a VAR model with fewer inputs than outputs is estimated): Its one-step nature makes it potentially more efficient and the same dynamics might be modeled with fewer parameters. The right MFD realization allows for more generality compared to the singular VAR since the covariance of the $N$-dimensional common factors $\chi_{t}$ at lag $0$, $\mathbb{E}\chi_{t}\chi_{t}'$, can be of rank $r<N$. Moreover, the rank of the spectral density of the common factors is $q<r$. We analyse the properties of the column Kronecker canonical form (which can be written as a minimal state space realization), and highlight its usefulness for modeling the common factor of GDFMs. Using the state space representation of the right MFD, estimation can be performed efficiently via Kalman filtering.